On the derivation of Fourier's law for coupled anharmonic oscillators

We study the Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice and subjected to a stochastic forcing mimicking heat baths of temperatures T_1 and T_2 on the hyperplanes at x_1=0 and N. We introduce a truncation of the Hopf equations describing the stationary state of the system which leads to a nonlinear equation for the two-point stationary correlation functions. We prove that these equations have a unique solution which, for N large, is approximately a local equlibrium state satisfying Fourier law that relates the heat current to a local temperature gradient. The temperature exhibits a nonlinear profile.